Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

More documents

Recommendations

Info

H&B – p.24/27Deduction cycloidSnell’s law between successive layerssinα jv j= sinα j+1v j+1, j = 1, 2,... ,NAs N → ∞ this gives another conservation law ∗sinαv= C (2)From conservation laws (1) en (2) will obtainparametrized curveα ↦→ (x(α),h(α))∗ Think of translation symmetry in x–direction and Noether’s theorem
H&B – p.25/27A few more high school computationsLemma (abbreviating v ′ = dv/dh)v ′ = g vandcos α = Cv ′ dhdα ,Proof: Differentiate (1) with respect to h and (2) with respect to αQEDThuswhile:dxdαIntegration then givesdhdαlemma== tan αdhdα1Cgv cos α(2)= 12C 2 g(3)= v sin α (2)= 12C 2 (1 − cos2α)gsin2α (3)h(α) = h 0 − 14C 2 g cos2αx(α) = x 0 + 14C 2 (2α − sin2α) :gcycloid with rolling angle ϕ = 2α and radius ̺ = 1/(4C 2 g)QED
Page 1 and 2: H&B - p.1/27Huygens and Bernoulli
Page 3 and 4: H&B - p.3/27Huygens and BernoulliCh
Page 6 and 7: H&B - p.6/27Commentary1. Potential
Page 8 and 9: H&B - p.8/27Arclength cycloidarclen
Page 10: H&B - p.10/27Evolute and evolventSo
Page 14 and 15: The master himself . . .H&B - p.14/
Page 16 and 17: H&B - p.16/27Detailed formulation1.
Page 18 and 19: H&B - p.18/27Johann Bernoulli’s s
Page 20 and 21: H&B - p.20/27DiscretisationHorizont
Page 22 and 23: H&B - p.22/27OpticsPn 1 = 1/v 1α 1
Page 26 and 27: H&B - p.26/27Commentary1. Challenge

Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

Create successful ePaper yourself

Delete template?

Save as template?